The proof of Lemma 3 is given at the end of this appendix.
The result in part (a) is well-known and versions of
it can be found in [1]}, [2]},
and [3]}, among others.
Result (b) in a convex-concave special case (\(x\)  and \(h\)  convex in \(\gamma \) 
and concave in \(\delta \) ) has been obtained by [4]}.
Our main interest is in part (c), the proof of which
relies on parts (a) and (b). The finiteness of \(\Gamma _{0}(x_{0})\)  is assumed
in part (c) because the mapping taking functions \(x\)  in \(l^{\infty }(\Gamma \times \Delta )\) 
to functions \(\sup _{\delta \in \Delta }x(\gamma ,\delta )\) , \(\gamma \in \Gamma \) ,
in \(l^{\infty }(\Gamma )\)  is in general not Hadamard directionally
differentiable without some additional assumptions.
